Osaka, the reigning US and Australian Open champion, has seven career hard-court titles, but has never managed to lift a clay- court trophy.
A leg injury meant Osaka missed last season's clay-court swing, and her last individual tournament on the surface was at Roland Garros in 2019.
Barty, who is assured of retaining her No 1 ranking next week, will play for the title Saturday against the winner of the second semi-final between No 8 Bianca Andreescu and No 23 Maria Sakkari.
Seventh-seeded Spaniard Roberto Bautista Agut beat Medvedev 6-4, 6-2, for his third win in as many meetings with the Russian.
Miami Open: Maria Sakkari ends Naomi Osaka's winning streak to reach semi-finals; Andreescu advances
Osaka broke in the second game of the second set and seized a 3-0 lead before Sakkari roared back to seal a semi-final spot in the prestigious ATP Masters and WTA hardcourt tournament.
Miami Open: Naomi Osaka, Ashleigh Barty, Bianca Andreescu in quarter-finals; Stefanos Tsitsipas into last-16
The world's top-ranked player Ashleigh Barty has reached the Miami Open quarter-finals, getting there by holding off 14th-seeded Victoria Azarenka of Belarus 6-1, 1-6, 6-2
Qatar Open: Garbine Muruguza sets up Victoria Azarenka semi-final, Elina Svitolina, Karolina Pliskova eliminated
Azarenka, a two-time winner of the Australian Open in 2012 and 2013, reached the last four by seeing off Ukrainian top seed Elina Svitolina, 6-2, 6-4.
Qatar Open: Former champions Karolina Pliskova, Petra Kvitova, Victoria Azarenka through to quarters
It was Jabeur who defeated the Czech during their last encounter in Doha in 2020 but this time it was 2017 Qatar champion Pliskova who booked her ticket to her first hard-court quarter-final since 2020 Dubai.
'When you play in front of an empty big stadium, it just feels very strange,' said Novak Djokovic
Abu Dhabi Open: Aryna Sabalenka beats Maria Sakkari to set up final clash against Veronika Kudermetova
Sabalenka had eight aces in her 6-3, 6-2 victory over Maria Sakkari. Kudermetova defeated 18-year-old Marta Kostyuk 7-6 (8), 6-4.
Abu Dhabi Open: Veronika Kudermetova beats second seed Elina Svitolina; Maria Sakkari, Aryna Sabalenka into semis
Kenin lost to ninth-seeded Maria Sakkari 2-6, 6-2, 6-0 in the quarter-finals. Svitolina lost to unseeded Russian Veronika Kudermetova 5-7, 6-3, 7-6 (3).
Abu Dhabi Open: Sofia Kenin fights back to beat Yulia Putintseva; Maria Sakkari, Elina Svitolina through to quarters
Kenin posted 57 unforced errors to Putintseva's 43 but still prevailed 3-6, 7-6 (5), 6-4 over the 13th-seeded Putintseva.
Abu Dhabi Women’s Tennis Open: World No 292 Anastasia Gasanova stuns former top-ranked Karolina Pliskova
The 21-year-old Gasanova won 6-2, 6-4 to reach the round of 16 and record her first ever victory over a player in the top 100.
Kasatkina was the first winner of the new season, beating Wang Qiang 6-2, 3-6, 6-2 after trailing 2-0 in the deciding set. It was the Russian's first match since October, and Wang's first since March.
Ostrava Open: Aryna Sabalenka wins 12 straight games against qualifier Sara Sorribes Tormo in comeback
Aryna Sabalenka lost the first ten games before miraculously recovering to find the court with her groundstrokes and went on to take the next 12 games for a 0-6, 6-4, 6-0 win.
Taking place four months later than usual due to the coronavirus, the early summer warmth normally associated with the tournament has been replaced by a bone-chilling 10 degrees.
French Open 2020 Women's Singles preview: Favourable draw, spectacular form make Simona Halep firm favourite
Ahead of Roland Garros' return, Firstpost takes a look at the Women's Singles draw and attempts to determine the potential outcome of all four quarters
Williams, whose 23 major victories are an Open-era record, will have nothing but her own skill and will to get her through as she seeks a record-breaking seventh title on the hard courts at Flushing Meadows.
Western & Southern Open: Naomi Osaka withdraws from semi-final in protest, tournament 'paused' for a day
Naomi Osaka tweeted that as a Black woman, she feels compelled to pull out of the tournament to put a focus on police shooting Black people.
Serena Williams finished rather meekly in a 5-7, 7-6 (5), 6-1 upset against Maria Sakkari on Tuesday night at Western & Southern Open